What does the multiset monad functor do to morphisms?

Question

The multiset monad, $(M, \mu, \eta)$, on the category of sets takes a set, $A$, to $M(A)$, the set of every multiset on $A$. That is what $M$ does to the objects of SET, but what does $M$ do to the morphisms?

Answer 1

Given $f:S\to T$, $M(f)$ maps the multiset $\sum n_i s_i, n_i\in \mathbb N,s_i\in S$, to $\sum n_i f(s_i)$.